Finite Group Databasebeta

← #13167309

Subgroups — #13167309

Order 1728.

410 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723715C211yesmaximal, characteristic
2C3×#63982C211yesmaximal, characteristic
3#11723112C211yesmaximal, characteristic
4#11722779C211yesmaximal, characteristic, complemented
5#11722778C211yesmaximal, characteristic, complemented
6#11722775C211yesmaximal, characteristic, complemented
7#11722775C211yesmaximal, characteristic, complemented
8#10593501—31nomaximal
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C2×C3×#627C2211yescharacteristic
11C2×C3⋊#622C2211yescharacteristic
12C2×C3⋊#622C2211yescharacteristic
13C3×#3757—21no—
14C32×#217—21no—
15#88815C2211yescharacteristic
16#88815C2211yescharacteristic
17#88813—21no—
18C3×#3724C2211yescharacteristic
19C3×Dic3⋊Dic3—21no—
20C3×#878—61no—
21#63982D311yescharacteristic
22#63957—31no—
23#63955—61no—
24C22⋊#622—31no—
25#63743—31no—
26#63742—31no—
27#63740—31no—
28#63740—31no—
29C23×C33D411yescharacteristic
30C2×C3×C3⋊Dic3—21no—
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3C2311yescharacteristic
36C2×C32×Dic3—21no—
37C2×C32×Dic3—21no—
38C3×#627—21no—
39C3×#627—21no—
40C3⋊#622—21no—
41C3⋊#622—21no—
42#5045—91nomaximal
43C22×C3⋊Dic3C2×D311yescharacteristic
44C22×C3×Dic3—61no—
45C22×C3×Dic3—61no—
46C2×C3×Dic6—61no—
47C2×#627C2×D311yescharacteristic
48C2×#622—31no—
49C2×#622—31no—
50C32×C22⋊C4—61no—
51#3757—21no—
52C3×#217—21no—
53C3×#217—21no—
54C3×#217—41no—
55C3×#217—61no—
56C3×#217—61no—
57C3×#217—121no—
58C3×C4⋊Dic3—61no—
59C3×Dic3⋊C4—61no—
60C3×Dic3⋊C4—61no—
61#3724—31no—
62#3724—31no—
63#3724C2×D311yescharacteristic
64Dic3⋊Dic3—21no—
65Dic3⋊Dic3—61no—
66Dic3⋊Dic3—61no—
67Dic32—61no—
68Dic32—61no—
69Dic32—61no—
70C22×C33—21no—
71C22×C33—21no—
72C22×C33—21no—
73C22×C33C2×D411yescharacteristic
74C3×C3⋊Dic3—21no—
75C3×C3⋊Dic3—41no—
76C3×C3⋊Dic3—41no—
77C3×C3⋊Dic3#4111yescharacteristic
78C3×C3⋊Dic3#4111yescharacteristic
79C32×Dic3—41no—
80C32×Dic3—41no—
81C3×C22⋊Q8—91no—
82#926—91no—
83#878—61no—
84#877—181no—
85#835—91no—
86#835—91no—
87#830—91no—
88#808—91no—
89#807—91no—
90C23×C32—21no—
91C23×C32—21no—
92C23×C32—41no—
93C23×C32—41no—
94C23×C32#6711yescharacteristic
95C2×C32×C4—61no—
96C2×C32×C4—61no—
97C2×C3⋊Dic3—21no—
98C2×C3⋊Dic3—21no—
99C2×C3⋊Dic3—21no—
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3#6711yescharacteristic
102C2×C3⋊Dic3C22×D311yescharacteristic
103C2×C3×Dic3—21no—
104C2×C3×Dic3—21no—
105C2×C3×Dic3—21no—
106C2×C3×Dic3—21no—
107C2×C3×Dic3—41no—
108C2×C3×Dic3—41no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—121no—
120C2×C3×Dic3—121no—
121C2×C3×Dic3—121no—
122C2×C3×Dic3—121no—
123C3×Dic6—121no—
124C3×Dic6—121no—
125#627—21no—
126#627—21no—
127#622—61no—
128#622—61no—
129#485—271noSylow
130C2×C33—41no—
131C2×C33#12311yescharacteristic
132C2×C33#13511yescharacteristic
133C2×C33#13711yescharacteristic
134C2×C3×Q8—91no—
135C22×C3×C4—91no—
136C22×Dic3—61no—
137C22×Dic3—61no—
138C2×Dic6—61no—
139C3×C4⋊C4—91no—
140C3×C4⋊C4—181no—
141C3×C22⋊C4—61no—
142C3×C22⋊C4—61no—
143C3×C22⋊C4—121no—
144C3×C22⋊C4—181no—
145#217—21no—
146#217—61no—
147#217—61no—
148#217—121no—
149C4⋊Dic3—61no—
150C4⋊Dic3—91no—
151C4⋊Dic3—91no—
152Dic3⋊C4—61no—
153Dic3⋊C4—61no—
154Dic3⋊C4—181no—
155Dic3⋊C4—181no—
156C4×Dic3—181no—
157C4×Dic3—181no—
158C4×Dic3—181no—
159C2×C3⋊C8—91no—
160C2×C3⋊C8—91no—
161C22×C32—22no—
162C22×C32—22no—
163C22×C32—21no—
164C22×C32—21no—
165C22×C32—21no—
166C22×C32—21no—
167C22×C32—21no—
168C22×C32—41no—
169C22×C32—41no—
170C22×C32—41no—
171C22×C32—41no—
172C22×C32—41no—
173C22×C32—41no—
174C22×C32—42no—
175C22×C32—42no—
176C22×C32—81no—
177C22×C32—81no—
178C22×C32—81no—
179C22×C32—81no—
180C22×C32C2×#6711yescharacteristic
181C32×C4—121no—
182C32×C4—121no—
183C3⋊Dic3—21no—
184C3⋊Dic3—41no—
185C3⋊Dic3—41no—
186C3⋊Dic3#23711yescharacteristic
187C3⋊Dic3#23711yescharacteristic
188C3×Dic3—41no—
189C3×Dic3—41no—
190C3×Dic3—41no—
191C3×Dic3—41no—
192C3×Dic3—61no—
193C3×Dic3—61no—
194C3×Dic3—61no—
195C3×Dic3—61no—
196C3×Dic3—62no—
197C3×Dic3—81no—
198C3×Dic3—81no—
199C3×Dic3—121no—
200C3×Dic3—121no—
201C3×Dic3—121no—
202C3×Dic3—121no—
203C3×Dic3—121no—
204C3×Dic3—241no—
205C3×Dic3—241no—
206C22⋊Q8—91no—
207#117—271no—
208#107—271no—
209#106—271no—
210#103—271no—
211#103—271no—
212C22⋊C8—271no—
213C33#48511yescharacteristic, complemented, Sylow
214C23×C3—21no—
215C23×C3—21no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3GO+(4,𝔽2)11yescharacteristic
219C3×Q8—181no—
220C3×Q8—181no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4—61no—
225C2×C3×C4—91no—
226C2×C3×C4—91no—
227C2×C3×C4—121no—
228C2×C3×C4—121no—
229C2×C3×C4—181no—
230C2×C3×C4—181no—
231C2×C3×C4—181no—
232C2×C3×C4—181no—
233C2×Dic3—21no—
234C2×Dic3—21no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—62no—
240C2×Dic3—62no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243C2×Dic3—61no—
244C2×Dic3—61no—
245C2×Dic3—121no—
246C2×Dic3—121no—
247C2×Dic3—121no—
248C2×Dic3—121no—
249Dic6—121no—
250Dic6—121no—
251C3⋊C8—181no—
252C3⋊C8—181no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32—41no—
260C2×C32—41no—
261C2×C32—41no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—42no—
267C2×C32—42no—
268C2×C32—82no—
269C2×C32—82no—
270C2×C32#93511yescharacteristic
271C2×C32#94211yescharacteristic
272C2×C32#95011yescharacteristic
273C2×Q8—91no—
274C22×C4—91no—
275C2×C8—271no—
276C2×C8—271no—
277C4⋊C4—91no—
278C4⋊C4—181no—
279C4⋊C4—271no—
280C4⋊C4—271no—
281C22⋊C4—61no—
282C22⋊C4—181no—
283C42—541no—
284C22×C3—22no—
285C22×C3—22no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—42no—
296C22×C3—42no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—81no—
300C22×C3—81no—
301C22×C3—81no—
302C22×C3—81no—
303C22×C3C2×GO+(4,𝔽2)11yescharacteristic
304C3×C4—91no—
305C3×C4—91no—
306C3×C4—121no—
307C3×C4—121no—
308C3×C4—121no—
309C3×C4—121no—
310C3×C4—181no—
311C3×C4—241no—
312C3×C4—241no—
313C3×C4—361no—
314C3×C4—361no—
315Dic3—41no—
316Dic3—41no—
317Dic3—61no—
318Dic3—61no—
319Dic3—61no—
320Dic3—61no—
321Dic3—62no—
322Dic3—121no—
323Dic3—121no—
324Dic3—121no—
325Dic3—121no—
326Dic3—121no—
327Dic3—241no—
328Dic3—241no—
329C32—21no—
330C32—21no—
331C32—41no—
332C32—41no—
333C32#504511yescharacteristic, complemented
334C23D32⋊D311yescharacteristic
335Q8—181no—
336Q8—181no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—91no—
340C2×C4—91no—
341C2×C4—181no—
342C2×C4—181no—
343C2×C4—181no—
344C2×C4—181no—
345C8—541no—
346C8—541no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—42no—
354C2×C3—42no—
355C2×C3—41no—
356C2×C3—41no—
357C2×C3—41no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3—41no—
362C2×C3—82no—
363C2×C3—82no—
364C2×C3#6423311yescharacteristic
365C2×C3#6423711yescharacteristic
366C2×C3#6423911yescharacteristic
367C22—21no—
368C22—21no—
369C22—21no—
370C22C2×D32⋊D311yescharacteristic
371C4—91no—
372C4—91no—
373C4—121no—
374C4—121no—
375C4—181no—
376C4—361no—
377C4—361no—
378C3—21no—
379C3—21no—
380C3—41no—
381C3—41no—
382C3#1059350111yescharacteristic, complemented
383C2—41no—
384C2#1172431911yescharacteristic
385C2#1172432411yescharacteristic
386C2#1172432611yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, prime radicalC23×C33D4
socleC22×C33C2×D4
hypercenterC23D32⋊D3
hypocenterC33#485
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #11722779C2semidirect
5. #11722778C2semidirect
6. #11722775C2semidirect
7. #11722775C2semidirect
213. C33#485semidirect
333. C32#5045semidirect
382. C3#10593501semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#117237152
C3×#639822
#117231122
#117227792
#117227782
#117227752
#105935013
#50459